2003 AMC 12B Problem 8

Below is the professionally curated solution for Problem 8 of the 2003 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2003 AMC 12B solutions, or check the answer key.

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Concepts:digits casework

Difficulty rating: 1390

Let $\clubsuit(x)$ denote the sum of the digits of the positive integer $x.$ For example, $\clubsuit(8) = 8$ and $\clubsuit(123) = 1 + 2 + 3 = 6.$ For how many two-digit values of $x$ is $\clubsuit(\clubsuit(x)) = 3?$

$3$

$4$

$6$

$9$

$10$

Solution:

Let $y = \clubsuit(x).$ Since $x \le 99,$ we have $y \le 18.$ Then $\clubsuit(y) = 3$ requires $y = 3$ or $y = 12.$

The two-digit numbers with digit sum $3$ are $12, 21, 30$ ( $3$ values), and those with digit sum $12$ are $39, 48, 57, 66, 75, 84, 93$ ( $7$ values), for $10$ in all.

Thus, the correct answer is E.

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